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For the card question: 

It turns out that there really isn't a good strategy; you should guess whichever card to be the card with the highest number on it, so the best you can do is 1/3.

Reference: https://math.stackexchange.com/questions/3474127/what-is-the-best-betting-strategy-to-maximize-probability-of-winning/

I think this is a trick question.

\(a_{23}=a_{22+1}\\ so \\n=22\)

p=23/2*22=253 though

I work it out with the help of an AP sum formula.

40 people

c = # children                                 cost 11c

a = #adult  = 40 -c                          cost 20 (40-c)         summed = 728

11c       +       20(40-c)        = 728      solve for c      then   a  = 40 - c

34 miles past the egg farm  +  12 miles from egg farm = 46 miles between the two trains

they are closing at  110 - 85 = 25 miles per hour

46 miles / 25 m/hr = 1.84 hrs when they meet

Notice that when we divide e into halves and rearrange those halves, the shape is the same as the shaded area. The shaded area is 1/8 of the total square(try dividing the diagram into only the gray area shape). We know the area of the square is 16, and that e is equivalent to 1/8 the area so its area is 1/8 * 16= 2 square inches.

Wouldn't   f^-1(x)  have to be given a valid domain in order for it to be a function?

So    f( f^-1(x) ) =  f(x)  would only be strictly true for a given domain ?

But yes,  I guess that is what they are after.

PQ =  9√3 / tan(30º)

AB = 12        BD = 6

AD = sqrt[( AB)²- ( BD)²]

BE = sqrt[( BD)²+(AD/2)²]

A right triangle has two sides of length 6 units each. What is the area of the triangle, in square units?

y = -5/3 x + 6      switch the x and y  then solve for y

x = -5/3 y + 6

y = - 3/5  (x-6)

You can find the answer here:::   https://web2.0calc.com/questions/please-help_66642#r2

Ohhh! Thank you! You helped me realize what I kept doing wrong. Thanks!

I got 18 and it turned out correct. 

AREA_{righttriangle} = 1/2 b * h   = 1/2  *  6 * 6 = .............. units²

Sorry! The latex didn't show up completely, so the question was incorrect. I changed it to a different question!

There is no c in the problem.

Sorry...I don't understand...

The image below with two squares, ABCD and BFGE, sharing a vertex. Given that AE = 5, what is the length of DG?

AB = 12         BE = 8          AX = 4

tan 30 =  opposite side / adjacent side

1/sqrt3  = 9 sqrt3 / PQ         SOLVE for PQ            (  1/sqrt3  is tan 30 )

Here is one way to find the quadratic

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)   =  (7+- sqrt23i)/2        so  a  =1   and b = -7

x^2 -7x + c              to find c      sqrt(  -7^2-4(1)(c))  = sqrt 23i        

                                  solving for c = 18

x^2-7x+18 =0                            

Hint:  f( f^-1(a) )  =  a.

Use sin:  sin(39)  =  x/10

Because you did not include a picture, I don't know which of the 16 triangles are shaded.

Use the formula  Area  =  ½·pi·radius²  to find the areas of the following semi-circles.

Step 1)  Find the area of the largest semi-circle; it would be grey if it were competely

                     shaded and its radius is 8.

Step 2)  Find the area of the next-largest semi-circle; it would be white if it were completely

                     shaded and its radius is 6.

Step 3)  Find the area of the second-smallest semi-circle; it would be grey if it were completely

                     shaded and its radius is 4.

Step 4)  Find the area of the smallest semi-circle; it would be white if it were completly

                     shaded and its radius is 2.

Start with the answer of Step 1; subtract the area of Step 2; add the area of Step 3; and, finally,

                     subtract the area of Step 4.